Lie algebra cohomology and the fusion rules

We prove a vanishing theorem for Lie algebra cohomology which constitutes a loop group analogue of Kostant's Lie algebra version of the Borel-Weil-Bott theorem. Consider a complex semi-simple Lie algebra and an integrable, irreducible, negative energy representation of. Givenn distinct pointsz _k in , with a finite-dimensional irreducible representationV _k of assigned to each, the Lie algebra of-valued polynomials acts on eachV _k , via evaluation atz _k . Then, the relative Lie algebra cohomologyH ^* is concentrated in one degree. As an application, based on an idea of G. Segal's, we prove that a certain homolorphic induction map from representations ofG to representations ofLG at a given level takes the ordinary tensor product into the fusion product. This result had been conjectured by R. Bott.     

On the Casimir operators associated with any Lie algebra

We give a characterization of the Casimir operators of a Lie algebra by polynomial solutions of a system of first-order partial differential equations. Further an upper bound is stated for the number of independent Casimir operators of a nilpotent Lie algebra.     

Quasifinite highest weight modules over the Lie algebra of differential operators on the circle

We classify positive energy representations with finite degeneracies of the Lie algebraW ₁₊ and construct them in terms of representation theory of the Lie algebra of infinites matrices with finite number of non-zero diagonals over the algebraR _m =[t]/(t ^m+1). The unitary ones are classified as well. Similar results are obtained for the sin-algebras.Supported in part by NSF grant DMS-9103792Supported in part by DOE grant DE-F602-88ER25066     

On the local Chevalley cohomology of the dynamical Lie algebra of a symplectic manifold

We describe the relations between the local Chevalley cohomologies related to the adjoint representation of the Poisson Lie algebra of a symplectic manifold and the Lie algebras of all symplectic or globally Hamiltonian vector fields of the manifold. The proofs are based on the computation of the cohomology of the complex (E, ), where E is the space of multilinear local maps from a vector bundle of a manifold M into the space of forms on M and L=d L.     

A Lie algebra structure on variation vector fields along curves in 2-dimensional space forms

A Lie algebra structure on variation vector fields along an immersed curve in a 2-dimensional real space form is investigated. This Lie algebra particularized to plane curves is the cornerstone in order to define a Hamiltonian structure for plane curve motions. The Hamiltonian form and the integrability of the planar filament equation are finally discussed from this point of view.     

Fock representations and BRST cohomology inSL(2) current algebra

We investigate the structure of the Fock modules overA ₁ ⁽¹⁾ introduced by Wakimoto. We show that irreducible highest weight modules arise as degree zero cohomology groups in a BRST-like complex of Fock modules. Chiral primary fields are constructed as BRST invariant operators acting on Fock modules. As a result, we obtain a free field representation of correlation functions of theSU(2) WZW model on the plane and on the torus. We also consider representations of fractional level arising in Polyakov's 2D quantum gravity. Finally, we give a geometrical, Borel-Weil-like interpretation of the Wakimoto construction.     

The algebra of differential operators on the circle and W KP (q)

Radul has recently introduced a map from the Lie algebra of differential operators on the circle of W _n . In this Letter, we extend this map to W _KP ^(q) , a recently introduced one-parameter deformation of W_KP - the second Hamiltonian structure of the KP hierarchy. We use this to give a short proof that W is the algebra of additional symmetries of the KP equation.     

On the isotropy subalgebras of Lie algebras of conformal vector fields

The conformal isotropy algebra of a point m in an n-manifold with a metric of arbitrary signature is shown to be locally reducible, by a conformal change of the metric, to a homothetic algebra near m iff, by choice of a chart, its constituent vector fields are simultaneously linearisable at m and, for n≥3, a necessary and sufficient condition for this in terms of the first and second derivatives of these fields at m is given. The implications for the Riemannian case and the Lorentzian case are investigated. In contrast to the former, a Lorentzian manifold admitting a conformal vector field that is not linearisable at some point need not be conformally flat. Relevant four-dimensional examples are provided.     

Finite dimensional representations of theSU(2)-current Lie algebra

We give a complete classification of the finite dimensional solutions for the Lie functional equations ofSU(2).     

Dynamical Lie algebra and conditional operators for many fermion Green functions

A method is introduced to calculate thermodynamic Green functions. A powerful theorem by Masson is the motivation for expressing the Fourier-time transform of the Green function as a super space matrix element of the resolvent of the Liouville operator. The eigenvalues of the Liouville operator are then expressed in a form first suggested by Judd for atomic systems and which are shown to be members of a Lie group.     

Commutative rings of partial differential operators and Lie algebras

We give examples of finite gap Schrödinger operators in the two-dimensional case.     

Cyclic homology of differential operators,the Virasoro algebra and aq-analogue

We show how methods from cyclic homology give easily an explicit 2-cocycle on the Lie algebra of differential operators of the circle such that restricts to the cocycle defining the Virasoro algebra. The same methods yield also aq-analogue of as well as an infinite family of linearly independent cocycles arising when the complex parameterq is a root of unity. We use an algebra ofq-difference operators andq-analogues of Koszul and the Rham complexes to construct these quantum cocycles.     

Invariant differential operators for non-compact Lie groups: Summary of su(4,4) multiplets

The present paper is part of the project of systematic construction of invariant differential operators of noncompact semisimple Lie algebras. Here we give a summary of all multiplets containing physically relevant representations including the minimal ones for the algebra su(4, 4). Due to the recently established parabolic relations the results are valid also for the algebras sl(8, R) and su*(8)     

The spectrum of the algebra of skew differential operators and the irreducible representations of the quantum Heisenberg algebra

The left spectrum of a wide class of the algebras of skew differential operators is described. As a sequence, we determine and classify all the algebraically irreducible representations of the quantum Heisenberg algebra over an arbitrary field.     

On the physical representations of an infinite-dimensional Lie algebra

Irreducible Hermitian representations of the infinite-dimensional Lie algebra, presented by J. Formánek [Czech. J. Phys.B16 (1966), 1,281] and suitable for the classification of elementary particles, are characterized by assumptions I–III. All representations of this kind are found explicitly.     

Matrix canonical realizations of the Lie algebra o(m,n). II. Casimir operators

The matrix canonical realizations of the Lie algebra of pseudo-orthogonal group O(m, n) described in the first part of this paper are further investigated. The explicit formulae for values of the Casimir operators (which are multiples of identity in these realizations) are obtained.     

Vector fields and differential operators: Noncommutative case

A notion of Cartan pairs as an analogy of vector fields in the realm of noncommutative geometry has been proposed in Czech. J. Phys. 46 (1996), p. 1197. In this paper we give an outline of the construction of a noncommutative analogy of the algebra of differential operators as well as its (algebraic) Fock space realization. We shall also discuss co-universal vector fields and covariant derivatives.     

On the determination of elliptic differential and finite difference operators in vector bundles overS 1

For an elliptic differential operatorA overS ¹, , withA _k (x) in END(^r) and as a principal angle, the -regularized determinant Det A is computed in terms of the monodromy mapP _A , associated toA and some invariant expressed in terms ofA _n andA _n–1 . A similar formula holds for finite difference operators. A number of applications and implications are given. In particular we present a formula for the signature ofA whenA is self adjoint and show that the determinant ofA is the limit of a sequence of computable expressions involving determinants of difference approximation ofA.Partially supported by an NSF grant     

On the algebra of symmetries of Laplace and Dirac operators

We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai–Ito algebra.